Hopf algebras and invariants of the Johnson cokernel

Abstract

We show that if $H$ is a cocommutative Hopf algebra, then there is a natural action of $Aut\left(F_n\right)$ on $H^{\otimes n}$ which induces an $Out\left(F_n\right)$ action on a quotient $\overline{H^{\otimes n}}$. In the case when $H=T\left(V\right)$ is the tensor algebra, we show that the invariant ${Tr}^C$ of the cokernel of the Johnson homomorphism studied in Algebr. Geom. Topol. 15 (2015) 801–821 projects to take values in $H^{vcd}\left(Out\left(F_n\right);\overline{H^{\otimes n}}\right)$. We analyze the $n=2$ case, getting large families of obstructions generalizing the abelianization obstructions of Geom. Dedicata 176 (2015) 345–374.